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Calculation of Shear Areas and Torsional Constant using the Boundary Element Method with Scada Pro
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3. Calculation of shear areas
Beam element subjected to transverse forces develops a shear strain, which is almost
always associated with flexural strain. In case that the direction of the externally imposed
transverse forces passes through the shear center of the cross-section of the beam, shear
strain is developed exclusively (absence of torsion), as the shear center is the point of the
cross-section of which the internally developed shear force passes through (shear stresses
integral).
(a)
(b)
Img. 3.1. Warping due to shear force for rectangular (a) and square hollow (b) section.
In general cases, in the cross-section of the beam, shear stresses caused by shear forces
are developed nonuniformly. Thus, the distribution of the shear deformation will be
nonuniform, which forces the cross-section to shear warping along the longitudinal
direction, i.e. Bernoulli’s acceptance rule (plane cross-sections remain plane and orthogonal
to the deformed beam axis after flexural) can no longer be assumed (
Euler-Bernoulli
flexural beam theory
) (Img.3.1).
If the shear force is constant along the axis of the beam and the longitudinal
displacements causing the warping are not restrained, the applied shear load is undertaken
only by shear stresses which are maximized at the boundary of the cross-section. This type
of shear is called
uniform
. Conversely, if the shear force varies along the beam and/or the
shear warping is restrained by load or support conditions the shear stress is developed
nonuniformly and shear is called
nonuniform
.
The warping due to shear is generally small compared to the corresponding due to
torsion, thus the warping stresses due to shear can be reasonably ignored in the analysis.
Thus, the stress field of the beam due to shear force is usually determined considering
uniform shearing, while the displacement field including shear warping is taken into
account indirectly through appropriate shear correction factors. Timoshenko (1921) was the
first to take into account the influence of the shearing deformation through shear correction
factors
, by suitably modifying the equilibrium equations of the beam. For that reason
the beam theory that takes into account the influence of shear deformations is also known as
Timoshenko flexural beam theory
. Note that the inverse of the shear correction factor
is